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A008919 Numbers n such that n written backwards is a nontrivial multiple of n. 10
1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, 10891089, 10999989, 21782178, 21999978, 108901089, 109999989, 217802178, 219999978, 1089001089, 1098910989, 1099999989, 2178002178, 2197821978, 2199999978, 10890001089 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

There are 2*Fibonacci([(n-2)/2]) terms with n digits (this is essentially twice A103609). - N. J. A. Sloane, Mar 20 2013

REFERENCES

W. W. R. Ball and H. S. M. Coxeter. Mathematical Recreations and Essays (1939, page 13); 13th ed. New York: Dover, pp. 14-15, 1987.

Martin Beech, [Title?], Mathematical Gazette, 74 (No. 467, March 1990), 50-51.

G. H. Hardy, A Mathematician's Apology (Cambridge Univ. Press, 1940, reprinted 2000), pp. 104-105 (describes this problem as having "nothing in [it] which appeals much to a mathematician.").

Benjamin V. Holt, A Determination of Symmetric Palintiples, arXiv:1410.2356, 2014

L. H. Kendrick, Young Graphs: 1089 et al, arXiv:1410.0106, 2014

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..400

Patrick De Geest, Palindromic Products of Integers and their Reversals

D. J. Hoey, Palintiples

D. J. Hoey, Palintiples [Cached copy]

Benjamin V. Holt, Some General Results and Open Questions on Palintiple Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 14, Paper A42, 2014.

Leonard F. Klosinski and Dennis C. Smolarski, On the Reversing of Digits, Math. Mag., 42 (1969), 208-210.

Lara Pudwell, Digit Reversal Without Apology, Mathematics Magazine, Vol. 80 (2007), pp. 129-132.

N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.

R. Webster and G. Williams, On the Trail of Reverse Divisors: 1089 and All that Follow, Mathematical Spectrum, Applied Probability Trust, Sheffield, Vol. 45, No. 3, 2012/2013, pp. 96 - 102.

Eric Weisstein's World of Mathematics, Reversal

FORMULA

If reverse(n) = k*n in base 10, then k = 1, 4 or 9 [Klosinski and Smolarski]. Hence A008919 is the union of A001232 and A008918. - David W. Wilson

MAPLE

P:=proc(q) local a, b, n; for n from 1 to q do a:=n; b:=0;

while a>0 do b:=b*10+(a mod 10); a:=trunc(a/10); od;

if type(b/n, integer) then if b/n>1 then print(n); fi; fi;

od; end: P(10^10); # Paolo P. Lava, Jul 29 2014

MATHEMATICA

Reap[ Do[ If[ Reverse[ IntegerDigits[n]] == IntegerDigits[4*n], Print[n]; Sow[n]]; If[ Reverse[ IntegerDigits[n + 11]] == IntegerDigits[9*(n + 11)], Print[n + 11]; Sow[n + 11]], {n, 78, 2*10^10, 100}]][[2, 1]] (* Jean-Fran├žois Alcover, Jun 19 2012, after David W. Wilson, assuming n congruent to 78 or 89 mod 100 *)

okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[ Flatten[ {99#, 198#}&/@Flatten[Table[FromDigits/@Select[Tuples[ {0, 1}, n], okQ], {n, 10}]]]] (* Harvey P. Dale, Jul 03 2013 *)

PROG

(Haskell)

a008919 n = a008919_list !! (n-1)

a008919_list = [x | x <- [1..],

                    let (x', m) = divMod (a004086 x) x, m == 0, x' > 1]

-- Reinhard Zumkeller, Feb 03 2012

CROSSREFS

Cf. A001232, A004086, A008918, A214927, A103609. Reversals are in A031877.

Sequence in context: A175698 A110819 A071685 * A110843 A023093 A001232

Adjacent sequences:  A008916 A008917 A008918 * A008920 A008921 A008922

KEYWORD

nonn,base,nice,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Corrected and extended by David W. Wilson Aug 15 1996, Dec 15 1997

STATUS

approved

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Last modified October 25 23:29 EDT 2014. Contains 248566 sequences.